Extended Cosmology in Palatini f(R)-theories
Paolo Pinto, Leonardo Del Vecchio, Lorenzo Fatibene, Marco Ferraris
EExtended Cosmology in Palatini f ( R ) -theories P. Pinto a, L.Del Vecchio b L.Fatibene b,c and M.Ferraris b a Physics Department, Lancaster University, Lancaster LA1 4YB, UK b Department of Mathematics, University of Torino (Italy) c INFN - Sezione TorinoE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We consider the cosmological models based on Palatini f ( R ) -theory for the func-tion f ( R ) = α R− β R − γ R , which, when only dust visible matter is considered, is called dunecosmology in view of the shape of the function f ( R ( a )) (being a the scale factor). We dis-cuss the meaning of solving the model, and interpret it according to the Ehlers-Pirani-Schildframework as defining a Weyl geometry on spacetime.Accordingly, we extend the definitions of luminosity distance, proper distance, and red-shift to Weyl geometries and fit the values of parameters to SNIa data. Since the theoreticalprediction is model-dependent, we argue that the fit is affected by an extra choice, namely amodel for atomic clocks, which, in principle, produces observable effects. To the best of ourknowledge, these effects have not being considered in the literature before. Corresponding author. a r X i v : . [ g r- q c ] F e b ontents We now have significant evidence that standard GR cannot account for observations if werestrict the sources of the gravitational field to the matter we are most familiar with. Thisis true now at cosmological scale (see [1], [2], [3], [4], [5]), for almost a century at the scalegalaxy cluster dynamics (see [6], [7]), and, more recently, at the scale of galaxies (see [8], [9],[10] and [11]).At more or less all scales, maybe with the exception of the Earth and the solar system,one can call for new sources we are not able to see directly yet (which are collectively named dark sources ) or, alternatively, for a modification of the dynamics of the gravitational field(i.e. the prescription of how ordinary visible matter generates a gravitational field). It is clearthat a priori the two approaches are equally possible (and somehow equivalent) until we havea direct evidence of dark sources other than gravitational ones, that, currently, we do nothave.In terms of modifications of dynamics, we have a (too) rich list of proposals: conformalgravity, MOND, metric, Palatini, and metric-affine f ( R ) -theories, torsion, Lovelock, just toquote a few. Each one has proven well in some specific situation, but none have yet provento be a solution in general. Hence, one needs a generic and robust framework to test modelswith a solid control on observational protocols, which often are not uniquely determined bythe action principle, but they are extra choices one does to connect to observations.For modifications of sources, in cosmology, the Λ CDM (-concordance) model has beenaccepted as a good description of current observations. Currently, observations require thecosmic pie to be Ω Λ (cid:39) . dark energy (in the form of a cosmological constant Λ ), Ω c (cid:39) . of some dark matter which we do not see though it shares with ordinary matter the sameequation of state (EoS) and of which we have no local, fundamental, direct evidence otherthan its gravitational effects, Ω b (cid:39) . of ordinary baryonic matter which accounts for the visible matter we see in form of galaxies and gas, as well as traces Ω r (cid:39) − of radiation (andrelativistic matter) which have very little effect today even though, due to a different scalingproperty, they grow important in an earlier universe. The Λ CDM (-concordance) model alsohas no spatial curvature k = 0 .The Λ CDM -model is considered the best description of current observations, a kind ofstandard which any other proposed model must reproduce. One also often believes that the– 1 –ark matter added from cosmological evidence is the same dark matter one needs in galaxyand cluster dynamics, an identification for which, however, a detailed quantitative parallelis missing and it is currently theoretically out of reach. To state the obvious, extrapolatingmatter equations of state (EoS), which are already an approximation, from a cosmologicalscale down to a galactic scale is brave, not even mentioning the lack of knowledge behind thefact we do not know an elementary counterpart for dark matter.Of course, the same bravery is needed in trying to extend the same dynamics from humanscale up to cosmology, not to mention down to the quantum world. Furthermore, often mostobservations rely on a gravitational model for interpretation and that such an interpretation isoften quite fragile with respect to modification of assumptions about geometry of spacetime.We have to admit that, beside raw data, we do not know much for sure and currently anystudy needs extra care.Among modifications of gravitational dynamics, we shall here consider a particular class,namely the extended theories of gravitation (see [12], [13], [14]). In extended theories of grav-itation, one has a Weyl geometry on spacetime, i.e. ( M, g, ˜Γ) , instead of the usual Lorentzianmetric structure. The (torsionless) connection ˜Γ is a priori independent of the metric g .While the connection describes the free fall of test particles (and light rays) in the gravita-tional field, the metric g is chosen to simplify the gravity-matter coupling and (consequently)to account for atomic clocks and, in turn, our protocols for measuring distances.The dynamics are chosen so that the metric and the connection, a priori independent,turn out to be a posteriori , i.e. as a consequence of field equations, EPS-compatible (see [15]),which means there exists a -form A = A (cid:15) dx (cid:15) such that ˜Γ αβµ = { g } αβµ − (cid:16) g α(cid:15) g βµ − δ α ( β δ (cid:15)µ ) (cid:17) A (cid:15) (1.1)This framework is theoretically inspired and motivated by a work on foundations ofgravitational physics by Ehlers-Pirani-Schild (EPS); see [15]. The application to extendedtheories is described in [16], [13], [14], [17].In an extended gravitational theory, one has modifications of dynamics which can equiv-alently be seen as effective sources. Extended theories also contain standard GR, with orwithout a cosmological constant, as a special (quite degenerate) case.A class of dynamics which are automatically extended theories of gravitation are Palatini f ( R ) -theories (see below and [18]), in which, field equations not only imply EPS-compatibility,but they also imply that A is closed or, equivalently, that ˜Γ is metric, i.e. ˜Γ = { ˜ g } for a metric ˜ g which is conformal to g . Being g and ˜ g conformal, they define the same pointwise causalstructure, the same light cones, the same light-like geodesics. However, they define differenttimelike geodesics so that it is important to declare that ˜ g , rather than g , defines the free fallof test particles. Also, being conformal, the g -length of a ˜Γ -parallelly transported vector isnot preserved, though at least it depends on the point only, not on the curve along which isparallelly transported.In this simpler case of two conformal metrics, the extra (kinematical) freedom one haswith respect to standard GR is encoded in the conformal factor ϕ , a scalar (real, positive)field such that ˜ g = ϕ · g . We have to stress, however, that, in view of the specific form of theaction functional in Palatini f ( R ) -theories, all these objects are not dynamically independent.For example, once we know the field ˜ g , then the conformal factor ϕ as well as the metric g are uniquely determined as function of ˜ g and its derivatives (up to order two). They arenot extra physical degrees of freedom, for example in the sense that one cannot “excite” one– 2 –ithout exciting the others. The dynamics of ϕ and g (or ˜ g ) are uniquely determined oncethe dynamics of ˜ g (or g ) is given, as it usually happens to Lagrangian multipliers. The metric g shares with Lagrangian multipliers the fact that it enters the gravitational Lagrangian withno derivatives (so, in a sense, its field equations are algebraic).Consequently, finding a solution in a Palatini f ( R ) -theory actually means determiningall ˜ g , the conformal factor ϕ , as well as the original metric g . Of course, one could recast theaction functional in terms of purely g or ˜ g , though at the price of making the matter–gravitycoupling more complicated and messing up with the interpretation of the theory about whichwe made a clear choice: times and distances are measured with g , free fall with ˜ g .Traditionally, doing all with the metric g is referred as the Jordan frame , while using ˜ g iscalled the Einstein frame . In the model we study in this paper, we do not use either the Jordanor the Einstein frame. We argue instead that one should not expect either of the metrics to beused for everything, as it happens in standard GR, and that is the essence of Weyl geometries.It is clear, for example by EPS kinematic analysis, that free fall and the causal structure arestructures coming from different physical phenomena (free fall is associated to test particles,causality to light rays) and one has no reason to assume a priori a constraint between them.The choice of ˜ g for geodesics and g for causal and metric structures is precisely what makesthis model different from the other analyses in which one frame is chosen to describe bothstructures. We shall eventually argue that this feature turns out to be in principle observableand important when one is going to compare the model with the solar system classical tests.The main aim of this paper is to discuss the application of a specific model of a Palatini f ( R ) -theory to cosmology. We discuss how the interpretation of the gravitational physics isextended from a Lorentzian metric geometry, to the more general Weyl conformal geometry,Furthermore, we investigate how the observations are interpreted in this more general setting(see [19]), which contains the standard GR case as a special case, as we said. If one doesnot like the model on a physical stance, one can regard this paper as a proposal for settinga rigorous standard for interpretation of observations in cosmology as well as an example ofhow a model should be discarded from an observational stance. As a matter of fact, Palatini f ( R ) -theories are naturally candidates to be at least a setting for understanding tests ofGR in a wider context, something which was originally done with Brans-Dicke theories forhistorical reasons (see [20]) while we are suggesting it should be done in extended gravity. We hereafter consider a cosmological model with a dynamics based on a Palatini framework,i.e. fundamental fields ( g µν , ˜Γ αβµ ) and an action functional A D ( g, ˜Γ , ψ ) = (cid:90) D (cid:18) √ g κ f ( R ) + L m ( g, ψ ) (cid:19) dσ κ := 8 πGc (2.1)where ψ denotes matter fields and √ gdσ is the volume element induced by the metric g . Thequantity √ g is the usual square root of the absolute value of the determinant of the metrictensor. The function f ( R ) will be here chosen as f ( R ) = α R − β R − γ R (2.2)where we set R := g µν ˜ R µν and ˜ R µν is the Ricci tensor of the (torsionless) connection ˜Γ .– 3 –he choice of function (2.2) has been not very thoughtful. If one sets γ = 0 the masterequation becomes too easy to be inverted, but one has no late acceleration. Moreover, herewe want to set up a framework able to deal with more generic f ( R ) , rather then choosingone for which we can easily do the computations. Under this viewpoint, the function (2.2)shows a number of pathologies which are good to learn to cope with, at least in a classicalregime. On the other hand, we do not want to add too many terms, keeping the degeneracyminimal. The more parameters one adds the easier the fitting becomes, the more experimentsone should use to actually remove degeneracy and to really constrain the parameters.If we had chosen γ = 0 , we would obtain a sort of Starobinsky model (see [21]), thoughin Palatini formalism. That is simpler to analyse (the master equation is globally and ana-lytically invertible). However, in this simplified model, one will have no negative pressure ineffective EoS, no late time acceleration, the conformal factor will be asymptotically constant.Still, also in this model, one has a bouncing rather than an initial singularity (in the Jordanframe).Similar models have been considered in the purely metric formulation, see [22] [23], [24],[25]. It has been argued that these models are not viable as physical models, mainly forstability issues and classical tests; see [27], [28], [29], [30]. On this basis, further models havebeen proposed in the metric formalism to address these shortcomings; see [31], [32], [33]. Evenif, in view of the non-equivalence between the metric and the Palatini formulation, the samecritiques do not apply directly to our model, we believe it is useful to briefly review them ina purely metric context to pinpoint what they exactly disprove.Most of the models proposed deal with cosmology only, where they used Jordan frame.All the critiques deal with stability and solar system tests (or, equivalently, Newtonian limit)in the Jordan frame as well. They show in many cases the Jordan frame is not viable in thesolar system. However, in cosmology, one uses only light and comoving test particles which,as a consequence of cosmological principle, are shared by the Jordan and Einstein frames.As a consequence, in cosmology (at least until perturbations are considered) one never reallyuses the metric ˜ g . If it is certainly true that solar system tests do not allow g -geodesics todescribe test particles, however, it is also true that assuming that test particles are describedby ˜ g does not change anything in the cosmological model and it gives a different solar systemmodel (which, by the way, can pass the tests quite easily, in view of the universality theorem;see [34]).Accordingly, we completely agree that (metric) f ( R ) -theories in the Jordan frame (atleast the ones considered in the references above) are not compatible with solar system tests.We just need to mention that when doing cosmology, one is not really compelled to declare theframe and there are mixed models, in which both g and ˜ g are used to do different things, whichhave not been discussed explicitly. It is our opinion they should be discussed and possiblydisproven as well. Let us stress that, in f ( R ) -theory there is no dynamical equivalencebetween purely metric and Palatini formalisms, even at the level of a simple counting ofphysical degrees of freedom. Accordingly, disproving metric models leaves Palatini modelsunchallenged.Of course, one could argue that, because of the form of the action functional, this modelis certainly non-renormalisable and that the Minkowski metric is not even a solution. Ofcourse, this is true also for standard GR with a cosmological term. It may be that themodel is not well suited for quantum gravity. However, on one hand, we do not know whatquantum gravity will eventually be precisely or whether it will require renormalisable theoriesor it will rather be non-perturbative in nature. On the other hand, we are here discussing a– 4 – lassical model, which unfortunately has nothing to do (observationally speaking) with thequantum regime. And there are many ways a classical model with a singular Lagrangiancan well behave, especially if the standard for well-behaviour is what happens in standardGR, in which singularities are already bound to appear generically. Kepler motion in a planeis another example of a singular Lagrangian which is accepted to describe a well-behaving(mechanical) system, in which the conservation of angular momentum prevents, most of thetimes, the system to get to the singularity.As far as the fact that Minkowski is not even a solution, the identification of Minkowski,and only Minkowski spacetime, as the vacuum state of gravitational field is already quitedubious at a fundamental level. A theory involving a metric field has no canonical vacuumjust because metrics (or vielbein) do not carry a linear (or affine) structure. Metric theoriesare different from all other fundamental field theories. Already standard GR is a peculiar fieldtheory in which one should learn to live without many of the structures used in field theoriesin Special Relativity (SR). For example, in GR one has no linear structure for configurations,generically no Killing vectors, no fixed background.In what follows, we shall assume that the connection ˜Γ is responsible for free fall. Par-ticles will follow geodesic trajectories of ˜Γ . The metric g is related to distances on spacetimeand its causal structures, e.g. the light cones. For example, a freely falling atomic clock willfollow a timelike geodesics worldline with respect to ˜ g , though the parameterisation is chosento be proper with respect to g . Of course, the difference is expected to be tiny, though wehave to keep in mind that we wish then to discuss objects going around for · s , withplenty of time to grow the tiny difference until it may become appreciated. Extrapolation atscales by many order of magnitudes requires good definitions and possibly no mathematicalapproximations.If experience still eventually points in favour of standard GR dynamics, we shall haveobtained it without relying on unnecessary theoretical assumptions, but based on experienceand a better understanding of which assumptions we rely on.In the literature, there are not many studies for Palatini f ( R ) -theories; see [35], [36]and references quoted therein. See also [37], [38], [39] for polynomial models. This is oftenargued to be due to a number of problems that Palatini f ( R ) -theories are supposed to havewhich have been however refuted; see [40], [41]. We shall not discuss here these issues sincethey are discussed in [42], [43], [44], [13], [45].Field equations for the action (2.1) are obtained by varying with respect to δg µν , δ ˜Γ αβν ,and δψ i : f (cid:48) ( R ) ˜ R µν − f ( R ) g µν = κT µν ˜ ∇ α ( √ gf (cid:48) ( R ) g βµ ) = 0 E i = 0 (2.3)In general, the second is solved by defining a conformal factor ϕ = ( f (cid:48) ( R )) m − , m beingthe dimension of spacetime, a conformal metric ˜ g µν = ϕg µν and by showing that ˜Γ = { ˜ g } isthence the general solution of the second field equation (which, written in terms of ˜ g and ˜Γ ,is actually algebraic, in fact linear, in ˜Γ ).The third equation E i = 0 is obtained as a variation of the action with respect to thematter fields ψ . It describes how matter fields evolve in the gravitational field. Usually, incosmology, one does not give a precise Lagrangian description of the matter dynamics, whichis described (under additional assumptions) by EoS, thanks to which Friedmann equationsbecome well-posed. Accordingly, we shall neglect the specific form for it.– 5 –y tracing the first equation by means of g µν , one obtains the so-called master equation f (cid:48) ( R ) R − m f ( R ) = κT (2.4)where we set T := g µν T µν . This is also an algebraic equation in R and T which genericallycan be (at least locally) solved for R = R ( T ) , so that the curvature R along solutions can beexpressed as a (model dependent but) fixed function of the matter content T .At this point, the first field equation can be recast as the Einstein equation for the metric ˜ g (or, equivalently, for the conformal metric g ) ˜ R µν −
12 ˜ R ˜ g µν = κ ˜ T µν ⇐⇒ R µν − Rg µν = κ ˆ T µν (2.5)In both cases, the energy–momentum stress tensors ( ˜ T µν or ˆ T µν ) need to be modified bysending to the right hand side all spurious contributions from matter (or curvature). Letus stress that also ˆ T µν differs from the original T µν which instead is the usual variation ofthe matter Lagrangian with respect to the metric δg µν . We shall not use ˆ T µν , while let usmention that ˜ T µν := 1 f (cid:48) ( R ) (cid:18) T µν − f (cid:48) ( R ) R − f ( R )2 κ g µν (cid:19) (2.6)This is where effective dark sources come from in Palatini f ( R ) -theories. Whatever visiblematter is, it is described by T µν , then ˜ T µν directly gets extra contributions from the modifieddynamics, i.e. from the function f ( R ) which, hopefully, by choosing it accordingly, can be usedto model dark matter and energy as effective sources. This is not the only effect in extendedtheories. Also the odd definition of atomic clocks (which are free falling with respect to ˜ g but proper with respect to g ) produces extra accelerations in particles. These accelerationsare universal, i.e. they are easily confused with an extra gravitational field acting on all testparticles equally which, when reviewed in a standard GR setting, calls for other sources.Hereafter, we shall investigate the combination of these two types of effects in cosmology.It is precisely because we chose ˜ g to describe test particles that we are not working inthe Jordan frame, and because we chose g to describe clocks that we are not working in theEinstein frame, either. That is true, even though, when we restrict to cosmology, one couldargue that we are working in the Jordan frame since the comoving structure is shared bythose two frames. However, the Einstein frame pops out again if we go to discuss gravity inthe solar system, where choosing the frame to describe test particles leads to different models. Let us consider a four dimensional spacetimes with a Weyl geometry ( M, g, { ˜ g } ) . If we wantto build a cosmological model based on the extended theories described above, we need toimpose the cosmological principle. Of course, with two metrics, one should at least stop andthink which metric should obey the cosmological principle. The good news is that (since themaster equation holds) it does not matter: g is spatially homogeneous and isotropic iff ˜ g is.The only difference is that, if g is in FLRW form in coordinate ( t, r, θ, φ ) , with a scale factor a , then ˜ g is in FLRW form in coordinate (˜ t, r, θ, φ ) , with a scale factor ˜ a = √ ϕ a . If theconformal factor is a function only of time, the new time is defined by d ˜ t = √ ϕdt .Thus one has a Friedmann equation both for a and ˜ a ˙ a = Φ( a ) ˙˜ a = ˜Φ(˜ a ) (3.1)– 6 –hich are, of course, defined to be equivalent. The specific form of the function Φ( a ) and ˜Φ(˜ a ) are obtained by expanding the Einstein equations (2.5), which are, in fact, equivalent.As a consequence of the cosmological principle, the energy-momentum tensor T µν is inthe form of a perfect fluid energy-momentum tensor, namely T µν = c − (cid:0) ( ρc + p ) u µ u ν + pg µν (cid:1) (3.2)for some time-like, future directed, g -unit, comoving vector u µ . Also ˆ T µν can be recast in thesame form (for different ˆ ρ and ˆ p ), as well as ˜ T µν is a perfect fluid energy-momentum tensorusing ˜ g and a suitable ˜ g -unit vector ˜ u as well as different effective pressure and density ˜ p and ˜ ρ . To establish an equivalence between Einstein equations and Friedmann equation, weneed conservation of the relevant energy-momentum tensors. Luckily enough, once again, if ∇ µ T µν = 0 (as it is, since it is variation of a covariant matter Lagrangian) then ˜ ∇ µ ˜ T µν = 0 and ∇ µ ˆ T µν = 0 as well. Here, ∇ µ denotes the covariant derivative with respect to g , ˜ ∇ µ thecovariant derivative with respect to ˜ g .Finally, we need to state the EoS for matter. This is where the game becomes odd:imposing the EoS for visible matter, i.e. for p and ρ appearing in T µν , the EoS for ˜ ρ and ˜ p in ˜ T µν are uniquely determined (as well as the EoS for ˆ ρ and ˆ p in ˆ T µν ). However, “simplicity” isnot preserved. Even if we assume visible matter to be simply dust (i.e. we select p = 0 as EoS)then the EoS for effective matter is determined, though the effective EoS is very exotic andnon-linear. Even if we regard it as a mixture of simple polytropic fluids, the decomposition isnot canonical and, in any event, it contains different polytropic fluids. Again, since we wish todiscuss a model at cosmological scale, it is necessary to avoid mathematical approximationsin EoS and learn to live with what we have, even when it is complicated to compute.As the usual in cosmology, one normalises the scale factor to be unit today, i.e. a ( t ) = 1 .Since the conformal factor is defined up to a constant factor which does not affect Christoffelsymbols { ˜ g } , one can also normalise the conformal factor to be (positive and) ϕ ( t ) = 1 , sothat the conformal transformation preserves the (positivity and) normalisation of the scalefactor and one has also ˜ a (˜ t ) = 1 .Regardless which equation of (3.1) we decide to solve, once we have t ( a ) , then the knowthe conformal factor as a function of a from the master equation, so we also have ˜ a ( a ) . Thenwe also know ϕ ( a ) and hence ˜ t ( a ) , ρ ( a ) , p ( a ) , ˜ ρ ( a ) , ˜ p ( a ) and so on. We get everything as afunction of a so all quantities are known in parametric form as function of every other (andno need to invert functions, other than the master equation).For an energy-momentum tensor in the form of a perfect fluid, we have T = c − (3 p − ρc ) and, for simplicity, we set the EoS for visible dust p = 0 . That, in view of energy-momentumtensor conservation, is equivalent to set ρ ( a ) := ρ a − .If we fix the function (2.2), the master equation reads as α R − β R + γ R − − (cid:18) α R − β R − γ R − (cid:19) = − α R + γ R − = κc (3 p − ρc ) (3.3)which can be solved in two branches (corresponding to the sign of R ) as ± R ( a ) = κ (cid:0) ρc − p (cid:1) ± (cid:113) κ ( ρc − p ) + 4 c αγ cα (3.4)– 7 –f we consider a mixture of dust ( p d = 0 ) and radiation ( p r = ρ r c ) ρc − p = ρ d c − p d + ρ r c − p r = ρ d c + ρ r c − ρ r c = ρ d c (3.5)Thus we have ρ d = ρ a − and, consequently ± R ( a ) = κc ρ d ± (cid:113) κ c ρ d + 4 c αγ cα = κcρ d ± (cid:113) κ c (cid:0) ρ d (cid:1) + 4 αγa αa (3.6)In case one wants different types of visible matter, though, the extra pressure wouldneed to be taken into account. We shall show the result for the values: α (cid:39) . β = 0 . m γ (cid:39) . · − m − (3.7)When we discuss fits in Section 5, we shall argue that if we analyse this model to fitSNIa there is a lot of degeneracy. The fit is not really able to constrain the parameters α and β , though if their values are provided by some other test (e.g. solar system tests) thensupernovae will fix γ . Still, the analysis of the model is interesting because having enoughtests to remove the degeneracy. At this point, we are introducing a small β and we are nothere concerned with the small value of α . See discussion in the conclusions.The conformal factor is chosen to be proportional to f (cid:48) ( R ) which is everywhere positiveif we use − R ( a ) , while + R ( a ) changes sign at about ρ := 1 . · kg m − . Thus, for theconformal factor to be positive, we need to define it in three branches • – the branch A, with R > and and ρ ∈ ( ρ , + ∞ ) (thus a ∈ (0 , a ) ), where theconformal factor is defined as ϕ A := − ϕ f (cid:48) ( + R ) ; • – the branch B, with R > and ρ ∈ (0 , ρ ) (thus a ∈ ( a , + ∞ ) ), where the conformalfactor is defined as ϕ B := ϕ f (cid:48) ( + R ) ; • – the branch C, with R < and ρ ∈ (0 , + ∞ ) (thus a ∈ (0 , + ∞ ) ), where the conformalfactor is defined as ϕ C := ϕ f (cid:48) ( − R ) ;where ϕ is a constant to be chosen so that today ϕ ( t ) = 1 .Branch A corresponds to very high densities, so it happened early in the universe. We as-sume then to currently be on branch B at a = a = 1 . So we choose ϕ := ( f (cid:48) ( + R ( a = 1))) − .The effective (mass) density and pressure are ˜ ρ = 4 γ R − β R + 12 κcρ R κc (3 α R − β R + γ ) ϕ ˜ p = − cγ R − cβ R − κp R κ (3 α R − β R + γ ) ϕ (3.8)where ρ and p are the total mass density and pressure of visible matter.If we have only visible dust, ρ = ρ d and p = 0 . If we have visible dust and radiation,then ρ = ρ d + ρ r , p = p r = ρ r c .We also clearly see that effective sources in general cannot be dust, as well as cannot bepolytropic.By using the correct expression for ϕ and R on each branch, we can compute theFriedmann equation ˙˜ a = κc ρ (˜ a ) ˜ a − kc =: ˜Φ(˜ a ) (3.9)– 8 –n view of the transformation between the two frames induced by the conformal factor,we have ˙ a = Φ( a ) := ϕ ( a ) (cid:18) d ˜ ada (cid:19) − ˜Φ( ˜ ρ ( a )) = ϕ ( a ) (cid:18) d ˜ ada (cid:19) − (cid:18) κc ρ ( a )˜ a ( a ) − kc (cid:19) (3.10)Hence, the Friedmann equation evaluated today reads as ω H = (cid:18) κc ρ ( ρ ) − kc (cid:19) (cid:18) ω := d ˜ ada (1) (cid:19) (3.11)Since the Hubble parameter today H is measured, then we can obtain the spatial curvatureas a function of the visible matter density ρ , i.e. k ( ρ ) = c − (cid:18) κc ρ ( ρ ) − ω H (cid:19) (3.12)Let us remark that once the function f ( R ) is fixed, then we know the function ˜ ρ ( ρ ) and theconstant ω .Accordingly, on each branch we can compute the function Φ( a ) , exactly, depending onthe parameters ( α, β, γ, ρ ) of the theory. In Figure 6.b in the Appendix we draw the graphof the function Φ( a ) for branches A and B.For any given value of the spatial curvature k one can compute the corresponding valueof the density which produces it. As usual the critical density is the density which producesa spatially flat spacetime k = 0 . Thus we have a 3-parameter family of extended models, allwith the observed value of the Hubble parameter today.One can explicit the time as a function of a solving the integral t ( a ) = (cid:90) a da (cid:112) Φ( a ) (3.13)Then the parametric curve γ : a (cid:55)→ ( t ( a ) , a ) represents the graph of the function a ( t ) . Letus notice that, in this way, one can study the function analytically, at the price of a finitenumber of numerical integration, even when the integral cannot be performed analytically.For realistic parameters, we obtain the evolution of the scale factor (dark-solid), com-pared with Λ CDM (light-solid) and standard GR (light-dashes).The 3 models are almost identical near today ( t = 0) while they differ in the past and inthe future. In particular, the extended f ( R ) model exhibits, for the critical density, a slightlyyounger age for the universe (with a Universe age of about . By ).Once we solved the model, then we can obtain all other quantities as a function of a .Qualitative graphs are collected in Appendix A. To fit data from supernovae (
SNIa ), we need a precise definition of the luminosity distance d L , the proper distance δ , and the red-shift z of a source within our model. In particular,we need to extend the standard discussion which is based on a Lorentzian geometry to an(integrable) Weyl geometry ( M, g, { ˜ g } ) .In cosmology, one defines spatial distances as the geometric distance on the surface t = t , without relying on synchronisation of clocks, which, of course, would be impractical in– 9 – igure 1 . Evolution of a ( t ) in standard GR for critical density (light dash), standard GR for ± of the critical density (light dotted), Λ CDM (light solid), and dune cosmology for the critical density(dark solid). Times in seconds ( x -axis), a is adimensional ( y -axis). astrophysics since easily it would take millions (if not billions) of years for a signal to bounceback and forth from another galaxy. Any comoving observer, at its time t = t , defines asurface t = t and chooses a geodesics on the surface (with respect to the metric inducedon the surface by g ). The proper distance δ ( t ) is then the g -length of such a (space-like)geodesics.Let us remark that the geodesic on the surface is in general not a geodesic on spacetime,as a geodesic on a sphere S in R is not a straight line.The surface t = t , which is here defined using the time coordinate, can also be definedin terms of the Killing algebra of isometries prescribed by the cosmological principle in anintrinsic fashion.The proper distance is a geometric well-defined distance, though it is difficult to definea protocol to measure it directly. We use it as a benchmark to refer the luminosity distance d L and the red-shift z of a source.In our model, light rays move along geodesics of ˜ g which are light-like with respect to g . However, since g and ˜ g are conformal, they share the same light-like geodesics. So we canconsider light-like geodesics using only the metric g . Accordingly, the red-shift is given by z ( a ) = a o − aa (4.1)where a e = a is the scale factor of g at emission, a o at observation. If we make observationstoday, of course we have a o = 1 . The scale factor ˜ a of the metric ˜ g does not play a role inobservations until we assume that atomic clocks are proper for g and not for ˜ g , though theyfree fall along ˜ g -geodesics.This can also be shown in detail, directly by repeating the standard argument in a Weylgeometry; see [46].Similarly, the area of the sphere S for t = t and r = r ∗ , measured by the metric g isexactly A = 4 πa ( t ) r ∗ . Accordingly, the luminosity distance of a comoving source at r = r ∗ observed at t = t is d L = (1 + z ) a o r ∗ (4.2)which, if k = 0 , is exactly d L = (1 + z ) δ ( t o , r ∗ ) . If one wishes not to assume k = 0 , dependingon the sign of k , r ∗ is anyway a known function of the proper distance.– 10 –here is no mathematical approximation we made here, in particular, we are not re-stricting ourselves to near sources by using a linear approximation of the Hubble law.Now we know z ( a ) as a function of the emission scale factor. If we observe today sourcesat different distances, we have their proper distance δ ( t o , r ∗ ) as a function of the emissionscale factor a δ ( a ) := c (cid:90) a a daa (cid:112) Φ( a ) (4.3)Given that we have d L ( a ) and z ( a ) , we have a parametric representation of the function d L ( z ) .That function depends on the dynamics of the model through the Weierstrass function,namely Φ( a ; α, β, γ, ρ ) , so that it brings information about the model, and the function f ( R ) in particular.For this reason, we can fit the parameters to obtain a best fit representation of theobserved curve. Since we know that the relationship between the magnitude of a far standard electromagneticsource (e.g. Ia type supernovae) and the observed redshift is determined by the dynamics ofthe scale factor a ( t ) , it is possible to use statistical inference methods to evaluate the agree-ment between the theoretical prediction within our model and the experimental observations.The complexity of the modelling of both the theory and observations requires correspond-ingly refined statistical and data analysis skills. In fact, the measurements of the magnitudeand the redshift of the SNIa must account for a strong uncertainty signal in the background(see [47]), usually described by two nuisance unknown parameters a and b . For this rea-son, a Bayesian inference approach is often used in this case (but more generally in all thecosmological measurements).We have decided to perform two different sets of fit for our model using the data con-cerning the measurement of the Ia type supernovae magnitude as a function of the observedredshift. We considered the Supernovae Legacy Survey (SNLS) project catalogue composedby 115 SNIa (see [47]) and the whole Union2.1 catalogue (see [48]) composed by 580 SNIa.We have a clear and unambiguous relation between these physical observables and the math-ematical objects in the theory, which makes these datasets particularly well suited for model-theoretic parameters fit. The SNLS is a smaller dataset than Union2.1 though its data aremore homogeneous, so we use it as a check for consistency.The theoretical value of the magnitude m of a source as a function of its redshift z hasbeen calculated considering the flow of power carried by the momentum-energy tensor asso-ciated to a high frequency electromagnetic wave propagating in a homogeneous and isotropicspacetime. One can notice that the explicit form of m ( z ) is strongly related to the dynamicsof the universe so it depends on the initial conditions (e.g. the baryonic and radiation energydensity today, the Hubble parameter today) as well as the vacuum Lagrangian parametersthat determine the evolution of metric tensor g .To fit the SNIa data, we relied on the software MULTINEST (see [49], [50], [51]), anefficient and robust Bayesian inference tool developed to calculate the evidence and obtain-ing posterior samples from distributions with (an unknown number of) multiple modes andpronounced (curving) degeneracies between parameters.The power of the software lies in the algorithm that naturally identifies the individualmodes of a distribution, allowing for the evaluation of the local evidence and parameter– 11 –onstraints associated with each mode separately. The fit was performed asking MULTINESTto find the free parameters of the theory ( α, β, γ, ρ ; a, b ) that minimizes the χ defined asfollowing: χ = N (cid:88) i =1 [ m Bi − m ( z (cid:63)i )] σ ( m B ) + σ int (5.1)where: • - z (cid:63)i is the measured redshift; • - m Bi = m i − a ( s −
1) + bc is the cleaned real magnitude in which: m i is the measuredmagnitude, s is the stretch factor, c is the colour factor, a and b are free nuisanceparameters that are fixed by the fit; • - m ( z (cid:63)i ) is the theoretical prediction of the magnitude at given redshift; • - σ ( m B ) = σ ( m ) + σ ( s ) + σ ( c ) is the error of the observations. In particular, wehave that σ ( m ) is the error related to the magnitude, σ ( s ) and σ ( c ) are respectivelythe errors of the stretch factor and color factor; • - σ int = 0 . the error related to the intrinsic dispersion of the real SNIa from theideal standard candle. • - N is the number of observed supernovae belonging to the dataset, 115 for the SNLSdataset, 580 for the Union2.1 dataset.MULTINEST is also able to provide us with the value of the χ evaluated on the bestfit parameters, the posteriors samples and the live points produced by the algorithm. Thisis very useful both for checking the right convergence to the minimum and to estimate theposterior probability distribution of our parameters. The posteriors analysis as well as theconfidence region has been performed using the software GetDist; see [52].Furthermore, considering the minimum value of the χ , we can compare different theo-retical models and determine the accuracy of the theoretical predictions.Different cases have been studied: at first, we have considered the case in which all theparameters ( α, β, γ, ρ , a, b ) are fitted against the Union2.1 dataset. That fit shows that themodel is strongly degenerate, that the β parameter does not influence the scale factor at thescales sampled by the dataset as well as that α is very poorly localized.Then we analysed cases in which some parameters are set to a value determined bydifferent considered scenarios. That shows that if we have some value for parameters fromtests other than SNIa, the model is still able to fit supernovae, for example to determine the γ parameter (as well as a and b ). This shows how SNIa are unable to fix all parameters ofthe model, as one can reasonably expect. The same behaviour in some sense is noticed instandard GR, where the parameters ( α, ρ ) are fixed at different scales, α being fixed by thesolar system tests, and ρ by supernovae.Here we expect something similar, only with more parameters. We expect different testsat different scales (solar system, light elements formation, . . . ) to remove the degeneracy andfix best fit value for the parameters. Then, and only then, the model can be tested to predictnew phenomena (e.g. BAO, lensing, . . . ). – 12 –he first fit we tried is using the Union2.1 dataset, fitting all parameters ( α, β, γ, ρ d , ρ r , H , a, b ) .At each step in the simulation, we compute the spatial curvature k using equation (3.12).The best fit parameters are α = 11 . +87 . − . β = − . +9 . − . · m M pc km − = − . +8 . − . · m γ = 2 . +8 . − . · − km m − M pc − = 0 . +0 . − . · − m − (5.2)The matter content is determined as ρ d = 1 . +1 . − . · − kg m − , ρ r = 2 . +0 . − . · − kg m − (5.3)the Hubble parameter as H = 73 . +1 . − . km s − M pc − = 2 . +0 . − . · − s − , (5.4)and the nuisance parameters as a = 0 . +0 . − . , b = 2 . +0 . − . (5.5)The spatial curvature is computed as k = − . +2 . − . · − m − (5.6)The fit has χ R = 0 . which could indicate an overestimation of the errors. See Figure 2 forthe triangular plot.The goodness of the fit is not particularly relevant here, since the best fit parametersconfidence intervals indicate that, as one could expect, the system is quite degenerate andsome of the parameters, α, β, γ in the first place, are quite poorly constrained. This fit is alsocompatible with negative densities of radiation which are of course unphysical. Let us noticethat, first of all, one can interpret it as saying that SNIa are not able to constrain radiation,i.e. that, as far as only SNIa are concerned, the radiation density could be zero. As forremoving degeneracy, one will need further tests to fix radiation contribution. Secondly, weare interested here in showing that the extended model we are discussing will be potentiallyable to fit data with the same visible sources and no dark sources at a fundamental level.This is a long process, it will take into consideration many different tests as explained inConclusions. In the following fits concerning SNIa, we shall fix a radiation density similarto the one predicted in the Λ CDM model, since SN themselves are not able to constrain thevalue of ρ r .The second fit we try is using Union2.1 dataset, fixing ρ d = 0 . · − kg m − , ρ r =0 . · − kg m − , and fitting all other parameters ( α, β, γ, H , a, b ) . At each step in thesimulation, we compute the spatial curvature k using equation (3.12).The best fit parameters are α = 0 . +99 . − . β = − . +17 . − . · m M pc km − = − . +16 . − . · m γ = 6 . +837 . − . · − km m − M pc − = 7 . +924 . − . · − m − (5.7)The Hubble parameter is determined as H = 73 . +1 . − . km s − M pc − = 2 . +0 . − . · − s − , (5.8)– 13 –nd the nuisance parameters as a = 0 . +0 . − . , b = 2 . +0 . − . (5.9)The spatial curvature is computed as k = − . +2 . − . · − m − (5.10)The fit has χ R = 0 . . See Figure 3 for the triangular plot.Also providing the densities for visible matter, the parameters of the model are stillpoorly constrained. Thus, as a third fit we fix α = 1 and β = 0 , as well as ρ r = 0 . · − kgm − . Then, still using the Union2.1 dataset, we fit all the other parameters ( ρ d , γ, H , a, b ) ,while still computing k at each step.The best fit parameter is γ = 2 . +3 . − . · − km m − M pc − = 2 . +4 . − . · − m − (5.11)The dust density and the Hubble parameter is determined as (cid:40) ρ d = 5 . +2 . − . · − kg m − H = 73 . +1 . − . km s − M pc − = 2 . +0 . − . · − s − (5.12)and the nuisance parameters as a = 0 . +0 . − . , b = 2 . +0 . − . (5.13)The spatial curvature is computed as k = − . +3 . − . · − m − (5.14)The fit has χ R = 0 . . See Figure 4 for the triangular plot.This time we see that the fit is better convincing. The only issue with respect to Λ CDM is that the dust density is (about one order of magnitude) higher than in Λ CDM . At thisstage it is not clear whether this is a prediction of the model or it is simply due to the valueimposed on the other parameters. For, let us consider a fourth fit, where we fix α = 0 . and β = 0 , as well as ρ r = 0 . · − kg m − . Then, always using Union2.1 dataset, we fitall other parameters ( ρ d , γ, H , a, b ) , still computing k at each step.The best fit parameter is γ = 2 . +3 . − . · − km m − M pc − = 2 . +3 . − . · − m − (5.15)The dust density and the Hubble parameter is determined as (cid:40) ρ d = 0 . − . − . · − kg m − H = 73 . +1 . − . km s − M pc − = 2 . +0 . − . · − s − (5.16)and the nuisance parameters as a = 0 . +0 . − . , b = 2 . +0 . − . (5.17)– 14 –he spatial curvature is computed as k = − . +3 . − . · − m − (5.18)The fit has χ R = 0 . . See Figure 5 for the triangular plot.In these cases the space curvature is about of the same order of magnitude of experi-mental constraints ( k ∼ − m − from Planck), just with a bigger uncertainty, as one canexpect since here we used only SNIa data. In any event, it is compatible with spatial flatness.We see that in this fourth fit the obtain more or less the same dust density as in Λ CDM ,thus confirming that one can change α and ρ d accordingly, so that the fit adjusts the valueof γ but still it is a good fit. That explicitly highlights the degeneracy we originally guessed.We also checked what happens in the third and fourth fits by restricting to the SNLSdataset. The SNLS is smaller, so we expect higher errors, though it is more homogeneous, sowe might prevent systematic errors in calibration. For this reason, it is interesting to checkthat one obtains about the same results as with Union2.1 dataset.We obtained best fit values which are completely compatible with the ones of the Union2.1 catalogue, both with reduced chi-squared values of χ R = 1 . .The last fits are reasonably good (maybe they show a overestimation of the errors). Theyshow us the way to test the model. One needs more tests, for example solar system testsand light elements formation, to fix α and β . Then, with that extra information, supernovaeseem to be able to determine the parameter γ (as well as the matter content ρ d , the Hubbleparameter H , and the nuisances a and b ).Let us stress that the value of γ is not compatible with γ = 0 showing that this modelis not a simple deformation of Starobinsky model f ( R ) = α R + β R . The model we areconsidering has late time acceleration, but supernovae do not constrain β which in some fitis even considered β = 0 .For the best fit values we found in the fourth fit, we showed the evolution of the scalefactor a ( t ) , see Figure 1. In Appendix A, we provide qualitative graphs for all relevantfunctions in the model. We can conclude that we have many degrees of freedom and degeneracy in the parameters,too many and too much to determine all parameters just by fitting the SNIa. However, if weimpose some theoretical constraints, for example by tests at different scales, we are alwaysable to determine at least the value of γ , as well as H , a and b , in different scenarios. Moretests are needed. Tests are model dependent and each new test must be done within a modelframework. It is dangerous and reckless to guess the results on the basis of an intuition whichhas been developed in standard GR.Only after one has enough tests to remove the degeneracy and fix the model, then newphenomena will allow us to discuss the predictivity of the theory. Currently, we believe thatsolar system test can fix α , while light elements production could fix β . Only at that pointcould one really discuss the physical content of the theory, for example by discussing BAO,CMB, structures formation, gravitational lensing, galaxies and clusters dynamics (see [53]),for each of which one still needs to develop a complete treatment within the framework of f ( R ) -Palatini theories.The specific model we consider here produces late time acceleration, it is compatible withSNIa observations (as many other models are, including Λ CDM ), it predicts a cosmological– 15 –ynamics which is quite closed to that of Λ CDM (see Figure 1), yet it is observationallydifferent in principle. That is simply not enough to propose or dismiss it as a sound physicalmodel.Even if we did not believe that the model f ( R ) = α R − β R − γ R − is a soundphysical proposal, it certainly shows that Palatini f ( R ) -theories are potentially able to modelSN observations and cosmology without adding dark sources at a fundamental level. Also,we showed that Palatini f ( R ) -theories are observationally different from standard GR and Λ CDM .This approach is also interesting because it poses a new standard for tests, with respectto the traditional comparison with Brans-Dicke models, which is now much better foundedon first principles, e.g. in view of EPS. Let us remark that if standard GR, in order to fitobservations, needs to account for dark sources (at all scales) at a fundamental level, thenalternative, modified or extended theories of gravitation need to account (thoroughly andin full detail) for how observations arise in their framework. One cannot simply say that amodel fits observations without dark sources, and not explaining in details where the extraaccelerations (which are usually interpreted as the effect of the gravitational field producedby dark matter) come from.Here we considered a specific Palatini f ( R ) -theory, we assumed an interpretation forthe fields g , ˜ g , and matter fields, which supports EPS framework and provides us with a solidbridge between the model and observational protocols. These assumptions are not proven,they are part of the model specification and can be falsified or corroborated by experiments.That is the correct way of proceeding.From our analysis, it seems that either α is much smaller than expected or dust density ismuch higher than usually supposed. Within this model,if a smaller value of α is not supportedby observations, then either one has more dust around or the model is contradicted by currentcosmological data. One could also argue that one needs the dark matter contribution to thedynamics ( Ω b +Ω c (cid:39) . ) for non-cosmological reasons (galaxies and cluster dynamics) whichare quite well established. However, one should say that to discuss galaxy models in a Palatini f ( R ) -theories, one should also consider the effect of conformal factor, that in these modelsis expected to depend on r , not (only) on cosmological t . Is that enough to fit observationswithout adding dark matter?We are also planning to devote future investigations to describe, the evolution of theHubble parameter H ( z ) as a function of distance (being the distance parameterised by z , a ,or t ) within this model. That will provide a test when the new data for Hubble drift willbe available and, at least, it provides new evidence that Palatini f ( R ) -theories can be, inprinciple, falsified by the observations.We are currently working to split the effects in extended theories, into the componentdue to effective sources from the effects due to the atomic clocks being proper with respectto a metric which is not the one describing free fall of test particles. Our interpretation iswell based in EPS framework, though being able to split the effects, will enable us to test thisassumption by experiments.Also a full analysis of the dynamical system describing the cosmological sector of thistheory would be interesting, as done in [39] for polynomial models. Here, we rather analysedthe system around a specific set of parameters, while a full analysis would highlight whetherone can have finite time singularities for other values of the parameters. For now, we knowno finite time singularity arises for our best fit parameters.Of course, one would need also perturbation theories, structure formation, lensing, in-– 16 –eractions with particle models (e.g. baryogenesis), the evolutions of perturbations of CMBand many other aspects. The point is that proceeding in that direction is the only way tofalsify models on a certain basis.And that is the only thing we can do, until we are able to discuss model propertieswithout fixing the function f ( R ) , something which currently is completely out of reach. Appendix A : Qualitative graphs The model is solved as soon as we get t ( a ) , see (3.13). All other quantities can be computedas functions of a so that they all can be plotted with respect to all the others in parametricform. For realistic parameters, graphs are difficult to be analyzed since the interesting featuresoccur at many order of magnitude away one from the others. So here we collect the graphsfor the parameters values κ = c = 1 α = 1 β = 116 γ = 1128 ρ = 34 (6.1)These graphs show qualitatively the form of model relations with realistic parameters.In view of the master equation and EoS for visible matter, equation (3.6) expresses thecurvature R as a function of the scale factor a . The function f ( R ) is given as a function of a as f ( + R ( a )) = 0 . − . a −
1) + O (( a − ) in Figure 6.a. That corresponds, neartoday, to approximately standard GR with a positive cosmological constant. Going back intime the action climbs the dune and falls down again unlike in Λ CDM . Also the limit for a −→ + ∞ is different from the standard case in which R ∝ a − . For this reason, we call thismodel dune cosmology .The Weierstrass function Φ( a ) , which determines the Friedmann equation for the scalefactor a , for these parameters is shown in Figure 6.b. One can see that there is a boundedallowed region (0 , a ] (corresponding to a universe which readily recollapses) as well as anunbounded one [ a , + ∞ ) in which we are now. The scale factor a = a acts as a reflectionpoint (i.e. a bouncing point). The second derivative of the Weierstrass function acts as adriving force. Accordingly, a positive second derivative indicates an accelerating expansionwhile around the maximum we have a deceleration phase.The conformal factor ϕ ( a ) is presented in Figure 6.c, ˜ a ( a ) = (cid:112) ϕ ( a ) a in Figure 7.a, andfinally ˜ t ( a ) in Figure 7.b, which is obtained by integrating the equation d ˜ tda = √ ϕ dtda = (cid:115) ϕ ( a )Φ( a ) (6.2)Then we integrate t ( a ) (in Figure 7.c) which in fact exhibits an initial slowing downphase, followed by an accelerated expansion.At that point one can graph all quantities as a function of all the others, for examplethe EoS for effective matter ˜ p ( ˜ ρ ) (see Figure 8.a) and the evolution of the scale factor of ˜ g asa function of ˜ t , i.e. ˜ a (˜ t ) (see Figure 8.b).As long as the old question of which frame is physical, if g or ˜ g , the issue is simply ignoredsince the quantities are all physical in a sense, just some physical structures are associated to g , some to ˜ g . The issue is also meaningless since the two metrics after all are one a functionof the other, so that they both equally are physical or unphysical in a sense.– 17 – cknowledgments We wish to thank N.Fornengo (University of Torino) and James Edholm (Lancaster Univer-sity) for comments and discussions. We also thank the (anonymous) referee for the stimulatingcomments about the first version of the manuscript.This article is based upon work from COST Action (CA15117 CANTATA), supported byCOST (European Cooperation in Science and Technology). We acknowledge the contributionof INFN (IS-QGSKY), the local research project
Metodi Geometrici in Fisica Matematica eApplicazioni (2017) of Dipartimento di Matematica of University of Torino (Italy). Thispaper is also supported by INdAM-GNFM.
References [1] A.G. Riess, et al.,
Observational Evidence from Supernovae for an Accelerating Universe and aCosmological Constant , In: AJ 116, (1998) 1009–1038; arXiv:astro-ph/9805201[2] F. Melchiorri, B.O. Melchiorri, L. Pietranera, B.O. Melchiorri,
Fluctuations in the microwavebackground at intermediate angular scales , The Astrophysical Journal. 250: L1, (1981).[3] E. Komatsu et al.,
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Cosmological Interpretation , 2009 ApJS 180 330; arXiv:0803.0547 [astro-ph][4] Planck Collaboration,
Planck 2015 results. XIII. Cosmological parameters , A&A 594, A13(2016); arXiv:1502.01589 [astro-ph.CO][5] The Fermi-LAT Collaboration,
The Fermi-LAT high-latitude Survey: Source CountDistributions and the Origin of the Extragalactic Diffuse Background , arXiv:1003.0895[astro-ph.CO][6] Zwicky, F.,
Die Rotverschiebung von extragalaktischen Nebeln , Helvetica Physica Acta, 6:(1933) 110?127[7] Zwicky, F.,
On the Masses of Nebulae and of Clusters of Nebulae , Astrophysical Journal, 86:(1937) 217[8] K.C. Freeman,
On the Disks of Spiral and S0 Galaxies , Astrophysical Journal 160, 811 (1970)[9] V.C. Rubin, J.W.K. Ford,
Rotation of the Andromeda Nebula from a Spectroscopic Survey ofEmission Regions , The Astrophysical Journal 159, 379 (1970).[10] V. Trimble,
Existence and nature of dark matter in the universe . Annual Review of Astronomyand Astrophysics. 25: (1987). 425–472.[11] J. de Swart, G. Bertone, J. van Dongen,
How Dark Matter Came to Matter , Nature Astronomy1, 0059 (2017)[12] L.Fatibene and M.Francaviglia,
Extended Theories of Gravitation and the Curvature of theUniverse – Do We Really Need Dark Matter? in : Open Questions in Cosmology, Edited byGonzalo J. Olmo, Intech (2012), ISBN 978-953-51-0880-1; DOI: 10.5772/52041[13] L.Fatibene, S.Garruto,
Extended Gravity , Int. J. Geom. Methods Mod. Phys., 11, 1460018(2014); arXiv:1403.7036 [gr-qc][14] Salvatore Capozziello, Mariafelicia F. De Laurentis, Lorenzo Fatibene, Marco Ferraris andSimon Garruto,
Extended Cosmologies , SIGMA 12 (2016), 006, 16 pages; arXiv:1509.08008[15] J.Ehlers, F.A.E.Pirani, A.Schild,
The Geometry of Free Fall and Light Propagation , in: GeneralRelativity, ed. L.O.‘Raifeartaigh (Clarendon, Oxford, 1972). – 18 –
16] M. Di Mauro, L. Fatibene, M.Ferraris, M.Francaviglia,
Further Extended Theories ofGravitation: Part I
Int. J. Geom. Methods Mod. Phys. Volume: 7, Issue: 5 (2010), pp.887-898; gr-qc/0911.2841[17] M.Roshan, F.Shojai,
Palatini f ( R ) gravity and Noether symmetry , Phys.Lett. B668 (2008)238-240; arXiv:0809.1272 [gr-qc] [18] S.Capozziello, M. De Laurentis (2015), F ( R ) theories of gravitation , Scholarpedia, 10(2):31422.[19] V.Perlick, Characterization of Standard Clocks by Means of Light Rays and Freely FallingParticles , Gen. Rel. Grav. (11), (1987) 1059-1073[20] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory ofRelativity
Wiley, New York (a.o.) (1972). XXVIII, 657 S. : graph. Darst.. ISBN: 0-471-92567-5.[21] A. A. Starobinsky,
A new type of isotropic cosmological models without singularity , Phys. Lett.B 91, (1980), 99-102[22] S. Nojiri, S.D. Odintsov and V.K. Oikonomou,
Modified Gravity Theories on a Nutshell:Inflation, Bounce and Late-time Evolution , Phys. Rept. (2017) 1doi:10.1016/j.physrep.2017.06.001 [arXiv:1705.11098 [gr-qc]][23] S.Nojiri, S.D. Odintsov,
Modified gravity with negative and positive powers of the curvature:Unification of the inflation and of the cosmic acceleration , Phys.Rev. D68 (2003) 123512; hep-th/0307288 [24] S.Nojiri, S.D. Odintsov,
Introduction to modified gravity and gravitational alternative for darkenergy , in: eConf C0602061 (2006) 06 , Int.J.Geom.Meth.Mod.Phys. 4 (2007) 115-146; hep-th/0601213 [25] S.Nojiri, S.D. Odintsov,
Unified cosmic history in modified gravity: from F(R) theory toLorentz non-invariant models , Phys.Rept. 505 (2011) 59-144; arXiv:1011.0544 [26] T.Chiba /R gravity and Scalar-Tensor Gravity , Phys.Lett.B575:1-3,2003;arXiv:astro-ph/0307338[27] L.Amendola, R.Gannouji, D.Polarski, S.Tsujikawa, Conditions for the cosmological viability of f ( R ) dark energy models Phys.Rev.D75:083504,2007 ; arXiv:gr-qc/0612180[28] L.Amendola, D. Polarski, SS.hinji Tsujikawa,
Are f ( R ) dark energy models cosmologicallyviable? Phys.Rev.Lett.98:131302,2007; arXiv:astro-ph/0603703[29] A.D. Dolgov, M. Kawasaki,
Can modified gravity explain accelerated cosmic expansion? ,Phys.Lett. B573 (2003) 1-4; arXiv:astro-ph/0307285[30] T.Chiba /R gravity and Scalar-Tensor Gravity , Phys.Lett.B575:1-3,2003;arXiv:astro-ph/0307338[31] A.A. Starobinsky, Disappearing cosmological constant in f ( R ) gravity , JETPLett.86:157-163,2007; arXiv:0706.2041 [astro-ph][32] S.A. Appleby and R.A. Battye, Do consistent F ( R ) models mimic General Relativity plus Λ ? Phys.Lett.B654:7-12,2007; arXiv:0705.3199 [astro-ph][33] W.Hu, I.Sawicki,?
Models of f(R) Cosmic Acceleration that Evade Solar-System Tests ,Phys.Rev.D76:064004,2007; arXiv:0705.1158 [astro-ph][34] A. Borowiec, M. Ferraris, M. Francaviglia, I. Volovich,
Universality of Einstein Equations forthe Ricci Squared Lagrangians , Class. Quantum Grav. 15, 43-55, 1998[35] A.Borowiec, M.Kamionka, A.Kurek, Marek Szydłowski,
Cosmic acceleration from modifiedgravity with Palatini formalism , arXiv:1109.3420 [gr-qc][36] G.J. Olmo,
Palatini Approach to Modified Gravity: f(R) Theories and Beyond , arXiv:1101.3864[gr-qc] – 19 –
37] M.Szydlowski, A.Stachowski, A.Borowiec,
Emergence of dynamical dark energy from polynomial f ( R ) -theory in Palatini formalism , Eur. Phys. J. C77, 603 (2017); arXiv:1707.01948 [gr-qc][38] M.Szydlowski, A.Stachowski, Simple cosmological model with inflation and late timesacceleration , Eur. Phys. J. C78, 249 (2018); arXiv:1708.04823 [gr-qc][39] M.Szydlowski, A.Stachowski,
Polynomial f ( R ) Palatini cosmology – dynamical systemapproach , Phys. Rev. D 97, 103524 (2018); arXiv:1712.00822 [gr-qc][40] E.Barausse, Thomas P.Sotiriou, J.C.Miller,
A no-go theorem for polytropic spheres in Palatini f ( R ) gravity , DOI: 10.1088/0264-9381/25/6/062001 (4th March 2008)[41] G. Magnano, L.M. Sokolowski, On Physical Equivalence between Nonlinear Gravity Theories
Phys.Rev. D50 (1994) 5039-5059; gr-qc/9312008[42] G.J.Olmo,
Re-examination of polytropic spheres in Palatini f ( R ) gravity , DOI:10.1103/PhysRevD.78.104026 (20th October 2008)[43] A. Mana, L.Fatibene, M.Ferraris A further study on Palatini f ( R ) -theories for polytropic stars ,JCAP 1510 (2015) 040 (2015-10-16) DOI: 10.1088/1475-7516/2015/10/040; arXiv:1505.06575[44] A. Wojnar, On stability of a neutron star system in Palatini gravity , Eur. Phys. J. C (2018) 78:421; arXiv:1712.01943 [gr-qc] [45] L. Fatibene, M.Francaviglia,
Mathematical Equivalence versus Physical Equivalence betweenExtended Theories of Gravitation , Int. J. Geom. Methods Mod. Phys. 11(1), 1450008 (2014);arXiv:1302.2938 [gr-qc][46] L.Fatibene,
Relativistic theories, gravitational theories and General Relativity , in preparation,draft version 1.0.0. https://sites.google.com/site/lorenzofatibene/my-links/libro-version-1-0-0/ [47] P. Astier, et al.,
The Supernova Legacy Survey: Measurement of Ω _ M , Ω Λ and w from theFirst Year Data Set , Astron.Astrophys.447:31-48,2006; arXiv:astro-ph/0510447[48] R. Amanullah et al., Spectra and Light Curves of Six Type Ia Supernovae at . < z < . and the Union2 Compilation , Astrophys.J.716:712-738,2010; arXiv:1004.1711 [astro-ph.CO][49] Farhan Feroz, M.P. Hobson, Multimodal nested sampling: an efficient and robust alternative toMCMC methods for astronomical data analysis , Mon. Not. Roy. Astron. Soc., 384, 2, 449-463(2008); arXiv:0704.3704[50] F. Feroz, M.P. Hobson, M. Bridges,
MultiNest: an efficient and robust Bayesian inference toolfor cosmology and particle physics , Mon. Not. Roy. Astron. Soc. 398: 1601-1614,2009;arXiv:0809.3437[51] F. Feroz, M.P. Hobson, E. Cameron, A.N. Pettitt,
Importance Nested Sampling and theMultiNest Algorithm ; arXiv:1306.2144[52] A. Lewis, S. Bridle,
GetDist , http://cosmologist.info/cosmomc/doc/programs/GetDist.htm and http://cosmologist.info/cosmomc/readme_gui.html . Accessed September 2016.[53] Jian-hua He, Luigi Guzzo, Baojiu Li, Carlton M. Baugh, No evidence for modifications ofgravity from galaxy motions on cosmological scales , Nature Astronomy (2018);arXiv:1809.09019. – 20 – .32 2.40 H
1e 18 b d
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1e 26 r
1e 26
Figure 2 . Triangular plot of the posterior distribution generated from the chains of MULTINEST.The coloured shapes represent the regions of the parameter space with a confidence level of σ and σ .We fit all parameters ( ρ d , ρ r , α, β, γ, H , a, b ) . We can see many parameters (e.g. α and β ) are notvery well constrained by the fit. H
1e 18 b a H
1e 18 b Figure 3 . Triangular plot of the posterior distribution generated from the chains of MULTINEST.The coloured shapes represent the regions of the parameter space with a confidence level of σ and σ .We fix ρ d and ρ r (to the density they have in Λ CDM for the sake of discussion) and fit ( α, β, γ, H , a, b ) .The values of α and β are still poorly constrained. – 21 – .32 2.40 H
1e 18 b d
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Figure 4 . Triangular plot of the posterior distribution generated from the chains of MULTINEST.The coloured shapes represent the regions of the parameter space with a confidence level of σ and σ .We now fix α = 1 , β = 0 , ρ r , fitting ( ρ d , γ, H , a, b ) . This time the constraint of values by the fittingis more convincing. H
1e 18 b d
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Figure 5 . Triangular plot of the posterior distribution generated from the chains of MULTINEST.The coloured shapes represent the regions of the parameter space with a confidence level of σ and σ .We now fix α = 0 . , β = 0 , ρ r , fitting ( ρ d , γ, H , a, b ) . This time the dust density is compatible with Λ CDM. – 22 – f ( + R ( a )) 1 a ( a ) 1 a' ( a ) 1 Figure 6 . The graphs show in order:a) the scalar density f ( + R ( a )) as function of the scale factor a b) the Weierstrass function Φ( a ) as function of the scale factor a c) the conformal factor ϕ ( a ) as function of the scale factor a a ˜ a ( a ) 1 a ˜ t ( a ) 1 ta ( t ) 1 Figure 7 . The graphs show in order:a) the scale factor ˜ a ( a ) of the metric ˜ g as function of the scale factor a b) the coordinate time ˜ t ( a ) as function of the scale factor a c) the evolution of the scale factor a ( t ) as a function of time t ˜ ⇢ ˜ p (˜ ⇢ ) 1 ˜ t ˜ a (˜ t ) 1 Figure 8 . The graphs show in order:a) the EoS ˜ p (˜ ρ ) of the effective sourcesb) the evolution of the scale factor ˜ a (˜ t ) of the metric ˜ g as a function of the coordinate time ˜ tt